Non-gray Model Research Articles

Efficient eigenvalue calculations in radiative transfer

An efficient method is used to compute the eigenvalues required in a discrete-ordinates solution to a special class of radiative-transfer problems. The basis for this computation is an algorithm for finding eigenvalues of a matrix that consists of the sum of a diagonal matrix and a rank-one matrix, a form that can arise in a discrete-ordinates solution of some basic transport problems. To illustrate the efficiency of the approach, a radiative-transfer problem relevant to a non-gray model with scattering that allows complete frequency redistribution is discussed.

Investigation of Entry Region Turbulent Flows of Non-Gray Gases

The entrance region turbulent flow between parallel planes has been investigated numerically. Variation of fluid properties with temperature has been considered. Monte Carlo simulation has been used with a narrow band non-gray model to investigate radiative transfer. The effects of radiative transfer and variable properties have been separately and systematically considered. Representative results are included for carbon monoxide and carbon dioxide. For the conditions considered, the results show that the variable properties exert a stronger influence compared to radiative transfer alone.

Synthetic Spectra and Mass Determination of the Brown Dwarf Gliese 229B

We present preliminary nongray model atmospheres and interiors for cool brown dwarfs. The resulting synthetic spectra are compared to available spectroscopic and photometric observations of the coolest brown dwarf yet discovered, Gl 229B (Nakajima et al.). Despite the grainless nature of the present models, we find that the resulting synthetic spectra provide an excellent fit to most of the spectral features of the brown dwarf. We confirm the presence of methane absorption and the substellar nature of Gl 229B. These preliminary models set an upper limit for the effective temperature of 1000 K. We also compute the evolution of brown dwarfs with solar composition and masses from 0.02 to 0.065 M☉. While uncertainties in the age of the system yield some indeterminateness for the mass of Gl 229B, the most likely solution is m ≈ 0.04-0.055 M☉. In any case, we can set an upper limit m = 0.065 M☉ for a very unlikely age, t = 10 Gyr.

Radiative heat transfer effects in chemically reacting nozzle flows

The two-dimensional spatially elliptic Navier-Stokes equations have been used to investigate the radiative effects in chemically reacting compressible flows of premixed hydrogen and air in an expanding nozzle. The radiative heat transfer is simulated using the Monte Carlo method. The nongray model employed is based on the statistical narrow-band model with an exponential-tailed inverse intensity distribution. The spectral correlation has been considered in the Monte Carlo formulations. Results obtained demonstrate that the radiative effects on the wall heat transfer may be significant. Extensive parametric studies are conducted to investigate the effects of equivalence ratio, wall temperature, inlet flow temperature, and the nozzle size on the radiative and convective heat fluxes on the walls.

Mass–Spectral Class Relationship for M Dwarfs

We derive mass-spectral class relationships for M dwarfs in the mass range 0.075 ≤ m/M☉ ≤ 0.6, which corresponds to a spectral range of M0-M10. The stellar models are based on the most recent nongray atmosphere models presently available. The results are in excellent agreement with the observed relationship derived by Kircpatrick & McCarthy for dwarfs of spectral type earlier than M7. We show that the spectral subclass is definitely not a linear function of mass and increases abruptly as the substellar mass limit is approached. This behavior reflects the observed trends in color, magnitude, and luminosity as a function of the mass and stems from the very physical properties of very low mass stars. Objects older than 1 Gyr with a spectral type earlier than M10 (for solar metallicity) are predicted to be main-sequence very low mass stars. For objects with such a spectral class to be bona fide brown dwarfs, they must be younger than 1 Gyr.

Mass-Luminosity Relationship and Lithium Depletion for Very Low Mass Stars

We derive mass-luminosity relationships for very low mass stars (0.06 < m/M☉ < 0.6) for different metallicities. Calculations are conducted with the different nongray atmosphere models presently available to illustrate the uncertainties in the stellar models. The theoretical mass-magnitude relation reproduces accurately the observed relationship in the V band (Henry & McCarthy) but still underestimates the flux in the K band by ~0.5 mag, a consequence of the still inaccurate water opacities. Calculations based on the gray approximation are shown to yield incorrect results over the entire mass range of interest. We also show that the fate of objects near the hydrogen-burning limit depends significantly on the treatment of the atmosphere. Nongray effects yield cooler and less luminous objects for a given mass near the stellar/substellar transition, and thus a lower hydrogen-burning minimum mass, m ≈ 0.07 M☉, for solar metallicity.

A SIMPLE APPROACH TO NON-GRAY GAS MODELING

Abstract A simple non-gray gas model for use in radiation computations is proposed. This is based on the idea of k distribution, or reordering the highly irregular variation of absorption coefficient to obtain a smooth function. In the present approach, a methodology to obtain the reordered distribution from a wide-band model is presented, which is simpler than previous methods. Several test problems, including nonisothermal and inhomogeneous media, are solved using the model. The validity of the approach is tested by comparing solutions to radiation problems with previously published results.

High heat flux in glass

Transient conduction and nongray radiation in an absorbing, emitting and isotropically scattering glass slab with reflective boundaries is solved by the application of an implicit finite difference/discrete ordinates method. Solutions are presented for predicting temperatures and heat fluxes in a one dimensional slab geometry being externally heated by a constant high heat flux at one boundary. The boundaries are assumed to be optically smooth with Fresnel reflection and have a uniform relative real refractive index. The window material is intensely heated to levels that exceed the softening/melting point temperature in order to categorized the transient onset of these conditions without phase or volume change. The numerical results reveal the effects of the system parameters on the heat transfer characteristics. Comparisons are made between a gray model and a 16 band, evenly spaced, nongray model in the 1 to 4 μ m wavelength region. Precise knowledge of the thermal behavior of the windows is essential to the development of glass and ceramic materials in diathermal and optical applications.

Investigation of Radiative Transfer in Nongray Gases Using a Narrow Band Model and Monte Carlo Simulation

The Monte Carlo method (MCM) is applied to analyze radiative heat transfer in nongray gases. The nongray model employed is based on the statistical narrow band model with an exponential-tailed inverse intensity distribution. The amount and transfer of the emitted radiative energy in a finite volume element within a medium are considered in an exact manner. The spectral correlation between transmittances of two different segments of the same path in a medium makes the statistical relationship different from the conventional relationship that only provides the noncorrelated results for nongray analysis. Two features of the MCM that are different from other nongray numerical methods are discussed. The simplicity of the MCM is demonstrated by considering the case of radiative transfer between two reflecting walls. The results for the radiative dissipation distributions and the net radiative wall heat fluxes are obtained for uniform, parabolic, and boundary layer type temperature profiles, as well as for a parabolic concentration profile. They are compared with available results of other methods. Good agreements are found for all the cases considered.

A combined heat-transfer analysis of a single-fiber CVD reactor

A combined-conjugated heat-transfer and fluid-flow analysis is presented for coating fibers by CVD in a vertical cylindrical quartz reactor. The numerical model focuses on radiation and natural convection. Three case studies are performed, and the wall temperature predictions are compared to experimental measurements. In the first case, the flowing gas is hydrogen, and conduction is more important than both radiation and convection, in which case measured and predicted wall temperatures agree excellently. In the second, hydrogen is replaced by argon, thus making radiation heat transfer more important than the previous situation. Three radiation models with increasing degrees of sophistication are compared: an approximate nongray model (no wavelength dependence of emissivity), an approximate semigray model, and a rigorous semigray model with view factor calculations. Comparison with experiments suggest that a semigray radiative analysis is needed for correct determination of wall temperatures. The third involves argon at a lower flow rate, where natural convection effects are more pronounced. Checking the validity of the Boussinesq approximation by incorporating the explicit dependence of density on temperature in the model shows a slight difference between the velocity fields predicted using the Boussinesq approximation and those obtained using the explicit dependence of density on temperature. However, there is negligible difference between the temperature fields predicted in the two cases.

Investigation of chemically reacting and radiating supersonic flow in channels

The two-dimensional time dependent Navier-Stokes equations are used to investigate supersonic flows undergoing finite rate chemical reaction and radiation interaction for a hydrogen-air system. The explicit multi-stage finite volume technique of Jameson is used to advance the governing equations in time until convergence is achieved. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The multidimensional radiative transfer equations for a nongray model are provided for general configuration, and then reduced for a planar geometry. Both pseudo-gray and nongray models are used to represent the absorption-emission characteristics of the participating species. The supersonic inviscid and viscous, nonreacting flows are solved by employing the finite volume technique of Jameson and the unsplit finite difference scheme of MacCormack to determine a convenient numerical procedure for the present study. The specific problem considered is of the flow in a channel with a 10° compression-expansion ramp. The calculated results are compared with the results of an upwind scheme and no significant differences are noted. The problem of chemically reacting and radiating flows are solved for the flow of premixed hydrogen-air through a channel with parallel boundaries, and a channel with a compression corner. Results obtained for specific conditions indicate that the radiative interaction can have a significant influence on the entire flowfield.

Transient energy transfer by conduction and radiation in nongray gases

A general formulation is presented to investigate the transient radiative interaction in nongray absorbing-emitting species between two parallel plates. Depending on the desired sophistication and accuracy, any nongray absorption model from the line-by-line models to the wide-band model correlations can be employed in the formulation to investigate the radiative interaction. Special attention is directed to investigate the radiative interaction in a system initially at a uniform reference temperature when the temperature of the bottom plate is reduced suddenly to a lower but constant temperature. The interaction is considered for the case of radiative equilibrium as well as for combined radiation and conduction. General as weU as limiting forms of the governing equations are presented, and solutions are obtained numerically by employing the method of variation of parameters. Specific results are obtained for CO, CO2, H2O, and OH. The information on species H2O and OH is of special interest for the proposed scramjet engine application. The results demonstrate the relative ability of different species for radiative interactions.

Radiation-Natural Convection Interactions in Two-Dimensional Complex Enclosures

A numerical finite-difference study has been carried out for the two-dimensional radiation-natural convection interaction phenomena in square enclosures with equal vertical finite-thickness partitions located at the centers of the ceiling and floor. Both participating gases (CO2 and NH3) and nonparticipating gas (air) are considered. In the radiation calculations, the nongray exponential wide-band models for CO2 and NH3 are used, together with a radial flux method utilizing a more realistic polar description for the radiation exchange in the enclosure. Results on the effects of both surface and gas radiation on the velocity and temperature fields and the overall heat transfer rates as functions of the partition heights at two levels of the Grashof number are presented and discussed in terms of the physical phenomena.

Effects of probe shape change on flow phenomena during Jovian entry

The effects of probe shape change on the flow phenomena around a Jovian entry body is investigated. The initial body shapes considered are: 45° sphere cone, 35° hyperboloid, and 45° ellipsoid. The radiating shock-layer flow is assumed to be axisymmetric, inviscid, and in chemical and local thermodynamic equilibrium. The radiative transfer is calculated with an existing nongray radiation model that accounts for molecular band, atomic line and continuum transitions. The results indicate that the shock-standoff distance, shock temperature and density, wall pressure distribution and radiative heating to the body are influenced significantly because of the probe shape change. The effect of shape change on radiative heating of the afterbody was considerably larger for the sphere cone and ellipsoid that for the hyperboloid. For the peak heating conditions, the net radiative heating to the body was found to be highest for the ellipsoid.

A Theoretical and Experimental Study of Radiation–Convection Interaction in a Diffusion Flame

An experimental and theoretical investigation of the interaction of gaseous thermal radiation with natural convection was made for a laminar methane-air diffusion flame in the lower stagnation region of a horizontal porous cylinder. The exponential wide-hand gas radiation model was employed in this nonhomogeneous (nonuniform in temperature and composition) problem through the use of scaling techniques. Using a numerical scheme, the compressible energy, flow, and species-diffusion equations were solved simultaneously with and without the radiative component. In the experiment, methane was blown uniformly from the surface of the porous cylinder, setting up (upon ignition) a diffusion flame within the free-convection boundary layer. Using a Mach-Zehnder interferometer and a gas chromatograph, temperature and composition measurements were obtained along the stagnation line. Excellent agreement was found between the results based on the nongray wide-band model and the experimental data. Furthermore, it was found that the wide-band model yielded results that were superior to those results that excluded radiation-interaction effects. Thus, this study demonstrates that the exponential wide-band model can be accurately applied to nonhomogeneous combustion situations in order to account for the radiation-convection interactions.

Radiative Viscous-Shock-Layer Solutions with Coupled Ablation Injection

This paper describes a viscous-shock-layer analysis for investigating high energy equilibrium flowfields about blunt axisymmetric bodies. The analysis includes radiation transfer, mass injection, diffusion, and visous effects. Radiation transfer is calculated with a nongray radiation model that accounts for line and continuum radiation. The analysis can treat large levels of blowing for both stagnation and downstream flow. Results are presented for selected Earth entry conditions that demonstrate the essential features of this analysis. Particular emphasis is given to the effects of mass injection on stagnation and downstream flow. Results for both specified and coupled ablation injection are presented.

SV CENTAURI - A TRULY CONTACT BINARY.

ABSTRACT A light-curve-synthesis program which implements interpolation in a grid of nongray model atmospheres has been used to determine the geometrical elements of SV Cen by simultaneous differentialcorrection fitting of the three-color light curves obtained in UBV by Irwin and Landolt (1972). Solutions with the mass ratio fixed at its spectroscopic value of 0.84 reveal an extraordinarily high degree of contact between the components (fill-out parameter f = 0.10 ::::: 0.02) and a rather high temperature of the less massive component relative to the more massive component. The large local temperature excess of the secondary star, X = A TIT = 0.78 :::: 0.03 (which results in temperatures 28,000 and 17,0000 K, for the less massive and more massive compounds, respectively) is difficult to understand in the framework of the contact model but might be related to the thermal-time-scale mass-transfer stage of the evolution of SV Cen. Key words: light-curve synthesis - mass transfer

The subsonic structure of radiatively driven winds of early-type stars

Methods are developed for the calculation of extended nongray model atmospheres for early-type stars with radiatively driven mass loss. Models for a central star of a planetary nebula and a Wolf-Rayet star are calculated, and compared with hydrostatic model calculations. Scaling laws are given for explaining the differences and similarities of stars of different stellar mass. It is found that transonic flow cannot occur if only continuum sources of opacity are present, because a moderate outward increase of the flux mean opacity is required for a physically acceptable structure. Observations of the Of star zeta Pup and the Wolf-Rayet star HD 50896 are summarized that indicate the continuum formation regions are spatially extended. (AIP)

Non-gray radiative transfer

The normal-mode-expansion technique is used to establish the solution of the Milne problem basic to a generalized equation of radiative transfer. The non-gray model used includes the effects of absorption, scattering and losses due to photo-electric ionizations and collisions of the second kind. Accurate numerical results are presented for such physical quantities as the extrapolation distance, the integrated Planck function and the angular distribution of the exit intensity for selected values of the basic parameters.

Radiative and convective heating during Venus entry.

Determination of the stagnation region heating of probes entering the Venusian atmosphere. Both convective and radiative heat-transfer rates are predicted, and account is taken of the important effects of radiative transport in the vehicle shock layer. A nongray radiative transport model is utilized which parallels a four-band treatment previously developed for air (Page et al., 1969), but includes two additional bands to account for the important CO(4+) molecular band system. Some comparisons are made between results for Venus entry and results for earth entry obtained using a viscous earth entry program.